1. Introductory Review of Cosmic Inflation
Shinji Tsujikawa (辻川信二 ) hep-ph/
Research Center for the Early Universe, University of Tokyo

2. Content Ingredients for standard big-bang cosmology What is inflation
Problems of the standard big-bang cosmology
Introducing the scalar field
Scalar field dynamics for inflation
Density perturbation and the origin of large scale structure
Reheating after inflation
Cosmic Inflation

3. Ingredients for Standard big-bang cosmology
Our tool: General Relativity
Based on the cosmological principle
Metric: Friedmann-Robertson-Walker metric
Energy-momentum tensor for perfect fluid:
Cosmic Inflation

4. Ingredients for Standard big-bang cosmology
Then the Einstein equations yield is the Planck energy hereafter =c=1
The equation of state for radiation p= for matter
When K=0 (flat infinite universe), the solution for radiation dominant: matter dominant:
In these simple cases the universe is expanding deceleratedly.
Scalar field
Horizon
Monopole
Cosmic Inflation

5. Ingredients for Standard big-bang cosmology
Introducing Hubble parameter and critical density
we can rewrite the Friedmann equation as
And we can further define , thus we have
Flatness Problem
Cosmic Inflation

6. What is “inflation”?
Inflation means a stage in the early universe, with accelerated expansion
It is also equivalent to r+3p<0 : negative pressure!
It is also equivalent to where the Hubble parameter
Without inflation, the standard big-bang cosmology would suffer from several severe problems. A.H.Guth first noted that introducing inflation would provide an efficient solution to these problems (A.H.Guth, Phys. Rev. D 23, 347, 1981).
Cosmic Inflation

7. Problems of the standard big-bang cosmology - Flatness problem
Friedmann equation can be rewritten as where In standard big-bang, for either radiation or matter dominant, always decreases.
is unstable: it tends to shift away from unity with the expansion of the universe
WMAP: , very close to one.
We require This is an extremely fine-tuning of initial conditions.
Cosmic Inflation

8. Problems of the standard big-bang cosmology - Flateness problem
With inflation: since
term increases during inflation, W rapidly approaches unity. As long as the inflationary expansion is sufficient, W stays of order unity even in the present epoch.
Cosmic Inflation

9. Problems of the standard big-bang cosmology - Horizon problem
Particle horizon: is the largest distance that can have casual contact at time t
Radiation dominant: matter dominant: therefore the horizon is much smaller in the past.
In fact the causally contacted surface of the last scattering surface only corresponds physical scale to the angle of order
Observationally, however, we see photons which thermalize to the same temperature horizon in all regions in the CMB sky.
Cosmic Inflation

10. Problems of the standard big-bang cosmology - Horizon problem
If there is an inflation period in the early universe, the scale factor a(t) would grow drastically, while the particle horizon would nearly stay unchanged. Then before inflation, the scale could be much smaller than horizon.
Therefore the isotropy in the CMB spectrum and the large scale structure can be solved.
Cosmic Inflation

11. Problems of the standard big-bang cosmology - Horizon problem
Another form of the Horizon problem: the origin of large scale structure.
Comoving wavelength: l. Physical wave lenghth: al
Perturbation scale larger than horizon can not be amplified and therefore can not form structure.
Larger scale perturbations enter the horizon later, and has less time to evolve, to form structure.
Cosmic Inflation

12. Problems of the standard big-bang cosmology - Horizon problem
Therefore it is practically impossible to generate a scale-invariant perturbation spectrum between the big bang and the time of the last scattering in the standard big-bang cosmology
COBE and WMAP have seen nearly scale-invariant perturbation spectrum .
WMAP:
Cosmic Inflation

13. Problems of the standard big-bang cosmology - Horizon problem
If there is inflation…
Early stage of inflation, the scale of perturbations is smaller than horizon, which form the seeds of large scale structure.
Perturbations grow out of horizon Scale of perturbation and are frozen.
After inflation: standard big-bang stage. The perturbations enter horizon horizon again. Then the frozen perturbations continue to evolve into structure.
spectrum
Cosmic Inflation

14. More about the origin of the large scale structure
Hubble radius: 1/H~t Comoving Hubble length: 1/aH.
Hubble radius (Hubble length) can be a good estimator of the particle horizon, both being ~ t
Later we shall not distinguish ‘horizon’ and ‘Hubble length’
Standard big-bang cosmology: aH always decreases, then comoving Hubble length increases all the time.
Inflation: aH increases, comoving Hubble length decreases.
Cosmic Inflation

15. More about the origin of the large scale structure
Early stage of inflation: l<1/aH, causality works to generate small quantum fluctuations, which form the seeds of large scale structure.
Then l>1/aH, perturbations are frozen
After inflation: standard big-bang stage. 1/aH increases. l<1/aH again, then causality works again. Then the frozen perturbations continue to evolve into structure.
The small perturbation imprinted during inflation appears as large-scale perturbations after this ‘the second horizon crossing’.
Cosmic Inflation

16. Problems of the standard big-bang cosmology-- Monopole problem
According to the view of particle physics, the breaking of supersymmetry (SUSY) leads to the production of many unwanted relics such as monopoles, cosmic strings, and topological defects.
String theory: gravitinos, Kaluza-Klein particle, etc.
Their energy density decrease as a matter component, much slower than radiation energy density. In radiation-dominant era, these massive relics could be the dominant materials in the universe, which contradicts with observations.
Cosmic Inflation

17. Problems of the standard big-bang cosmology-- Monopole problem
If there is inflation…
Provided that these unwanted relics are produced before inflation, their energy densities would decrease drastically with the fast increase of scale factor ‘a(t)’. Thus unwanted relics can be red-shifted away.
We still have to worry about those relics produced after inflation. Generally if the reheating temperature is sufficiently low ,the thermal production of unwanted relics, such as gravitinos, can be avoided.
Cosmic Inflation

18. Scalar fields in particle physics and cosmology
To obtain inflation, we need materials with the unusual property of a negative pressure.
It is normally imagined that inflation begins at the Planck scale. Therefore we have to seek for a quantum theory to describe the materials for inflation: scalar field (spin-0)
Because of Planck scale, it is most suitable to adopt a quantum theory of gravity to describe inflation.
Unfortunately this theory has not come yet.
Our approach is a semi-classical one: we do not quantize the gravity field. Quantum Field Theory + Classical background
Cosmic Inflation

19. Scalar fields in particle physics and cosmology
As yet, there has been no direct observation of a fundamental scalar particle (such as Higgs), but they play a crucial role in particle physics theory in bring about mass through spontaneous symmetry broken (SSB).
Earlier inflationary models simply use the Higgs field for the Grand Unified Theory (GUT) such as SU(5) and first order transitions.
However they can not meet the requirements of cosmology.
Cosmic Inflation

20. Scalar fields in particle physics and cosmology
In inflationary cosmology scalar fields are introduced in a more phenomenological way.
Anyway, we look for guidance about the likely form of the scalar field potential in particle theory, hoping that in the end, it will belong to a complete ‘Theory of Everything (TOE)’.
Recent trend is to construct inflationary models based on superstring or supergravity models.
The scalar field responsible for inflation is often called ‘inflaton’.
Cosmic Inflation

21. Scalar field dynamics
The standard way to specify a particle theory is via its lagrangian. The lagrangian of a single scalar field with potential V is
Then the energy-momentum tensor can be written as
Assuming the f is spatially homogeneous, or noting the fact that spatially gradient terms ~ , the energy-momentum tensor take the form of a perfect fluid with
Cosmic Inflation

22. Scalar field dynamics
Substituting the expression of r and p into the basic two equations, we have
During inflation we require r+3p<0, which yields Therefore a flat potential is required.
Once inflation gets under way, then the curvature term in the Friedmann equation becomes less and less important. Normally it is assumed negligible from the start.
Different inflationary models give different potentials.
Chaotic
Cosmic Inflation

23. Slow-roll approximation
spectrum
The standard technique for analyzing inflation is the slow-roll approximation:
Defining the so-called slow roll parameter one can verify that slow-roll approximation are valid when e<<1, |h|<<1
e and h are functions of V, therefore it is easy to see where inflation might occur. Inflation ends when e and h grow >~1
Cosmic Inflation

24. Relation between inflation and slow-roll
Slow-roll approximation is a sufficient condition for inflation. This can be qualitatively seen in the first approximated equation of motion.
Another way to see this explicitly: slow-roll requires consequently
One the other hand we have the definition of inflation is recovered.
Cosmic Inflation

25. Amount of inflation
We need sufficient amount of inflation to solve the flatness problem, horizon problem, ets.
Usually we define
To solve the flatness problem, we require N>~70
Similar value of N is required to solve the horizon problem.
Cosmic Inflation

26. A simple example – Chaotic inflation
This model is described by the quadratic of quartic potential
Substituting the form of V into previous equations, we have
We have a exponentially expanding solution:
Cosmic Inflation

27. A simple example – Chaotic inflation
The slow-roll parameter reads therefore the inflationary period ends around , after which the universe enters a reheating stage.
The total amount of inflation is approximately
In order to lead to sufficient inflation N>~70, we require the initial value to be
Cosmic Inflation

28. A simple example – Chaotic inflation
Detailed analysis should have more dependence on Quantum Field theory (QFT).
The detailed form of the inflation potential V should be corrected by loop correction in perturbation theory and renormalization theory.
The inflaton mass m can have dependence on f, whose form can be calculated from the Renormalization Group Equation (RGE).
Another important type of inflationary model is using multi-field, such as hybrid models of A.D.Linde (Phys. Lett. B, 259, 38, 1991)
Cosmic Inflation

29. Basic picture of density perturbation and the origin of large scale structure
At early stage of inflation, vacuum fluctuation of the inflaton field is generated. – Quantum Field Theory
After the first horizon crossing, the fluctuation grows as classical one, which forms the origin of large scale structure
After the second horizon crossing, the fluctuation evolve fully classically, which forms today’s universe.
Here I shall briefly show the first two stages: how quantum fluctuation forms the origin of large scale structure.
Inflationary models have most predictive power in this aspect. Therefore the observation can kill many inflationary models through observing the large scale structure.
Cosmic Inflation

30. Vacuum fluctuation in quantized scalar field
The lagrangian of a scalar field in arbitrary spacetime (metric) is
The field equation can be obtained by
For a flat spacetime, we adopt the Lorentz metric, then we have
For non-interacting, or free field, we have then the field equation is , namely the Klein-Gordon equation
Cosmic Inflation

31. Vacuum fluctuation in quantized scalar field
Inflaton: we need the Robertson-Walker metric instead of the Lorentz metric.
Then the field equation for the inflaton is split the field into an unperturbed part and a perturbation: f(x,t)=f(t)+df(x,t)
and given a Fourier component, we have
Cosmic Inflation

32. Vacuum fluctuation in quantized scalar field
After canonical quantization for , we have
where and are the annihilation and creation operator for an inflation with momentum k, respectively.
satisfies
Because annihilation operator annihilates the vacuum, we have the form of the vacuum fluctuation, which is purely a quantum effect.
Cosmic Inflation

33. Vacuum fluctuation in quantized scalar field
From slow-roll approximation, we can deduce that H varies very slowly during inflation.
Then we can seek a solution ignoring the variation of H:
Here L is the comoving box size for normalization. And k is the wavenumber. Remembering that 1/k and 1/aH means the comoving wavelength and the comoving Hubble length respectively, the epoch k=aH just means the crossing of horizon.
Cosmic Inflation

34. The spectrum of perturbation
A few Hubble times after horizon-crossing we have k<<aH therefore
The spectrum of density perturbation can be defined as
The spectrum of primordial curvature perturbation is given by
Because we are dealing with slow-roll inflation, H varies very slowly, the above expressions can be evaluated at the epoch of horizon exit k=aH
Cosmic Inflation

35. The spectrum of perturbation
Using slow-roll approximation, we can express as where e is the slow-roll parameter.
Now we can define the ‘effective spectral index’ n(k) as this is equivalent to the power-law behavior that is assumed when defining the spectral index in the normal way,
Cosmic Inflation

36. The spectrum of perturbation
A simple calculation evaluated at k=aH can show that
Because slow-roll requires e<<1 and h<<1, we draw an important conclusion: inflation predicts the spectrum is close to scale-invariant.
WMAP: strong support for the inflation scenario.
Cosmic Inflation

37. The spectrum of perturbation
Similarly we have
Different models have different e and h, therefore sufficiently precise measurement of the spectrum, n(k) and would discriminate different models.
WMAP:
Cosmic Inflation

38. Reheating: Recovering the Hot Big Bang
Reheating: the period of inflationary expansion gives way to the standard Hot Big Bang evolution
Typically reheating would have little impact on the predictions on density perturbation from the inflationary scenario.
However, reheating is crucial to our understanding of: whether baryogenesis can be brought about successfully; whether gravitinos might be over produced; whether topological defects can be produced after inflation.
Cosmic Inflation

39. Reheating: Recovering the Hot Big Bang
There are typically three periods of the reheating process 1. non-inflationary scalar field dynamics, 2. decay of inflation particles, 3. thermalization of decay product.
The theory of the second stage has recently gained important developments, which led to a significant change of view since books such as Kolb and Turner’s ‘The Early Universe’ (1990).
Cosmic Inflation

40. Reheating: Recovering the Hot Big Bang
Once inflation is over, slow-roll approximation is no longer valid. Recalling the general equation of motion for inflaton f:
The scalar field begins to oscillate about the minimum of the potential.
Then the equation of motion can be rewritten as the equation for the time-average energy density :
Cosmic Inflation

41. Reheating: Recovering the Hot Big Bang
If the particle decay is slow (e.g. if the only decay channels are into fermions), one can insert a phenomenological term directly into the above equation: such an equation can be used to describe the coherent oscillation of inflaton, slowly producing fermions
Recently it was found that the inflaton may decay into bosonic particles, allowing a decay by parametric resonance. This permits an extremely rapid decay of the inflaton particles.
This dramatically rapid decay has been termed preheating to distinguish it from the old scenario of reheating, which is now believed to happen later than preheating.
Cosmic Inflation

42. Reheating: Recovering the Hot Big Bang
The occupation number generated by parametric resonance (preheating) are huge, so that bosons created are far from thermal equilibrium.
Fermions and the Pauli exclusion principle.
Decay and thermalization: the bosonic particles produced in preheating should decay, interact, and finally reach thermal equilibrium. The details will be strongly dependent on the field theory adopted.
After reheating the universe is on its way of standard big-bang cosmology again.
Cosmic Inflation

43. Summary and discussion
Success of inflation: it solves a number of cosmological problems such as flatness, horizon, and monopole problems, and at the same time it generates the seed for nearly scale-invariant large scale structure.
Cosmological scenarios alternative to inflation: pre-big-bang (M.Gasperini and G.Veneziano, Astropart.Phys. 1, 317, 1993) and cyclic model (P.J.Steinhardt and N.Turok, Phys.Rev.D, 65, , 2002)
Problem: the origin of inflaton? What is the state of the universe before inflation?
Future: high-precision observation is expected to reveal the detailed nature of inflation; from theoretical side extensive works are consctructing viable models based on string and supergravity theories.
Cosmic Inflation

44. Recommendation for references
J.V.Narlikar and T.Padmanabhan, Gravity, Gauge theories, and Quantum Cosmology. D.Reidel, This book provides a careful introduction to the details of quantum cosmology. To do so the authors have described the ingredients of Gauge Field Theories and General Relativity before begin the discussion for quantum cosmology, including the inflationary scenario.
Cosmic Inflation

45. Recommendation for references
E.W.Kolb and M.S.Turner, The Early Universe, Addison-Wesley, This is classic book described ideas across the whole range of what had become known as particle cosmology or particle astrophysics, including such topics as topological defects, inflationary cosmology, dark matter, axions, and even quantum cosmology.
Cosmic Inflation

46. Recommendation for references
A.R.Liddle and D.H.Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, As a recent textbook, it provides a modern and unified overview of the inflationary scenario and the origin of density perturbation. Its discussion is very clear, and carefully compares predictions with observations.
Cosmic Inflation